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Cayley diagram

C3×C37⋊C3 #66527

order333
isoclinismisoclinic class of C37⋊C3
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×C37⋊C3
unambiguous semidirect productC37⋊C32
unambiguous semidirect product(C3×C37)⋊C3

Splittings

Fitting, socleC3×C37 ⋊ C3
derived, SylowC37 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3C37⋊C3
commutator, cosocle, hypocenterC37C32
FrattiniC1G
Fitting, socleC3×C37C3
LCSG ▷ C37
UCSC1 ◁ C3

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 111

A minimal faithful permutation representation — 2 generators on 40 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,21)(11,16,18)(13,19,15)(17,24,28)(20,26,25)(22,29,23)(27,32,33)(30,34,31)(35,37,36);(1,2,4,8,14,22,30,35,34,19,5,9,15,23,24,7,13,20,27,29,12,18,25,31,32,16,3,6,11,17,21,28,33,36,37,26,10)(38,39,40)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D37⋊C3, D3×C37⋊C3, C3⋊(D37⋊C3), …
maximal normal subgroup ofC3×D37⋊C3, D3×C37⋊C3, C3⋊(D37⋊C3), …
maximal quotient ofC2×C3×C37⋊C3, C9×C37⋊C3, #11758605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(333, 4);